Binomial distributions involve two choices — usually “success” or “fail” for an experiment. This binomial distribution calculator can help you solve **bimomial problems** without using tables or lengthy equations. You do need to know a couple of key items to plug into the calculator and then you’ll be set!

- Probability(P) — percentage or decimal
- Number of trials (n)
- Successes (X) — ranges are acceptable, for example an X of between 0 and 4 successes

- For the first box (p), enter the probability of success in a trial as a decimal. This may be given to you as a percentage (i.e. 80% of respondents…), or you may be given a word problem that you need to convert to a decimal (i.e. a multiple choice test with four answers would have a .25 probability of a right answer each time you guessed).
- In the second box, enter the number of trials (n).
- The next two boxes, X1 and X2, allow you to enter a range, i.e. from 0 to 4 you would enter 0 in the X1 box and 4 in the X2 box. If you do not want a range, but rather an exact number, enter the number twice–once in each box (i.e. for “exactly 9” you would enter 9 in both X1 and X2).

## Answer

The probability of between and successes is .

## How to find the answer

### The Way Mortal Humans Do It

If you’re like most people, using a formula over and over again to get the answers you want doesn’t sound like fun!

Most people use a binomial distribution table to look up the answer, like the one on this site. The problem with most tables, including the one here, is that it doesn’t cover all possible values of p, or n. So if you have p = .64 and n = 256, you probably won’t be able to simply look it up in a table.

The alternative method is to use a calculator like this one! Many scientific calculators like the TI-89 can find the answer to problems like these.

If you want to know how the numbers work, then read on!

### The “Mathy” Way

To figure out what the total probability is, first we have to figure out the probability of each value of *x*, using this formula:

n!
x!(n – x)! |
p^{x} (1-p)^{(n-x)} |

So if your range is from to , you’d have to use that formula for . Then when you got the answer from each of those, you’d add them all up together to get the total:

P( … ) =

The graph below shows each possible value of *x* along the bottom, and the bar represents the chance that *x* will actually equal that value during a real experiment. Yellow bars means the value is in the range you chose, and if you look at the list above, you’ll see the bars correspond to the answers, and you’ll also see that if you added up all the yellow areas, you’d get the total from above also.

If you prefer an online interactive environment to learn R and statistics, this *free R Tutorial by Datacamp* is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try *this Statistics with R track*.

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Can you assist with this QUESTION?

1. A survey of 80 families with 4 children has yielded the following results

Children [4 boys, 0 girls] [3 boys, 1 girl] [2 boys, 2 girls] [1 boy, 3 girls] [0 boys, 4 girls] [Total]

# of families 8 24 20 16 12 80

Do the results support the view that boys and girls are equally likely birth outcomes? Note this is a binomial distribution with p=1/2 ( since boy or girl is equally possible) and n = 4 since there are 4 children in each case so that will be 4 independent events. The relative frequencies for the 5 categories are 1 4 6 4 1.

Quite helpful.